We derive analytic formulas for fair strike prices of discretely-sampled ( forward-start) variance swaps under a class of stochastic volatility jump ( SVJ) models． This class covers a couple of stochastic volatility and jump models which have been studied widely in literature including both affine and non-affine models．We demonstrate a general methodology to find analytic formulas for the class while we obtain explicit solutions for several special cases． Numerical examples show that our solutions give close results to Monte Carlo simulations．Obviously，our explicit solutions beat the latter in speed．